C

Figure 1.1.9

1.1.6 Though we will see a simple proof of Theorem 1.1 in the next section, it is possible to prove it using methods similar to those in the proof of Theorem 1.2. Prove the special case of Theorem 1.1 where the points \(P = (x_{1}, y_{1}, z_{1})\) and \(Q = x_{2}, y_{2}, z_{2})\) satisfy the following conditions:
\(x_{2} > x_{1} > 0\), \(y_{2} > y_{1} > 0\), and \(z_{2} > 1 > 0\).
\(\textit{Hint: Think of Case 4 in the proof of Theorem (1.2), and consider Figure 1.1.9.}\)

Is there a scalar \(m\) such that \(m(\textbf{v} + 2\,\textbf{w}) = \textbf{k}\)? If so, find it.

1.2.2. For the vectors \(\textbf{v}\) and \(\textbf{w}\) from Exercise 1, is \(\norm{\textbf{v} - \textbf{w}} = \norm{\textbf{v}} - \norm{\textbf{w}}\)? If not, which quantity is larger?

1.2.3. For the vectors \(\textbf{v}\) and \(\textbf{w}\) from Exercise 1, is \(\norm{\textbf{v} + \textbf{w}} = \norm{\textbf{v}} + \norm{\textbf{w}}\)? If not, which quantity is larger?

B

1.2.4. Prove Theorem 1.5 (f) for \(\mathbb{R}^{3}\).

1.2.5. Prove Theorem 1.5 (g) for \(\mathbb{R}^{3}\).

C

1.2.6. We know that every vector in \(\mathbb{R}^{3}\) can be written as a scalar combination of the vectors \(\textbf{i}\), \(\textbf{j}\), and \(\textbf{k}\). Can every vector in \(\mathbb{R}^{3}\) be written as a scalar combination of just \(\textbf{i}\) and \(\textbf{j}\), i.e. for any vector \(\textbf{v}\) in \(\mathbb{R}^{3}\), are there scalars \(m\), \(n\) such that \(\textbf{v} = m\,\textbf{i} + n\,\textbf{j}\)? Justify your answer.

1.3.25. For nonzero vectors \(\textbf{v}\) and \(\textbf{w}\), the \(\textit{projection}\) of \(\textbf{v}\) onto \(\textbf{w}\) (sometimes written as \(proj_{\textbf{w}}\textbf{v}\)) is the vector \(\textbf{u}\) along the same line \(L\) as \(\textbf{w}\) whose terminal point is obtained by dropping a perpendicular line from the terminal point of \(\textbf{v}\) to \(L\) (see Figure 1.3.5). Show that

1.3.26. Let \(\alpha\), \(\beta\), and \(\gamma\) be the angles between a nonzero vector \(\textbf{v}\) in \(\mathbb{R}^{3}\) and the vectors \(\textbf{i}\), \(\textbf{j}\), and \(\textbf{k}\), respectively. Show that \(\cos^{2} \alpha + \cos^{2} \beta + \cos^{2} \gamma = 1\).

(Note: \(\alpha\), \(\beta\), \(\gamma\) are often called the \(\textit{direction angles}\) of \(\textbf{v}\), and \(\cos \alpha\), \(\cos \beta\), \(\cos \gamma\) are called the \(\textit{direction cosines}\).)

B

1.4.16. If \(\textbf{v}\) and \(\textbf{w}\) are unit vectors in \(\mathbb{R}^{3}\), under what condition(s) would \(\textbf{v} \times \textbf{w}\) also be a unit vector in \(\mathbb{R}^{3}\;\)? Justify your answer.

1.4.17. Show that if \(\textbf{v} \times \textbf{w} = \textbf{0}\) for all \(\textbf{w}\) in \(\mathbb{R}^{3}\), then \(\textbf{v} = \textbf{0}\).

C

1.4.26. Prove that in Example 1.8 the formula for the area of the triangle \(\triangle PQR\) yields the same value no matter which two adjacent sides are chosen. To do this, show that \(\frac{1}{2}\,\norm{\textbf{u} \times (-\textbf{w})} = \frac{1}{2}\,\norm{\textbf{v} \times \textbf{w}}\), where \(\textbf{u} = PR\), \(-\textbf{w} = PQ\), and \(\textbf{v} = QR\), \(\textbf{w} = QP\) as before. Similarly, show that \(\frac{1}{2}\,\norm{(-\textbf{u}) \times (-\textbf{v})} = \frac{1}{2}\,\norm{\textbf{v} \times \textbf{w}}\), where \(-\textbf{u} = RP\) and \(-\textbf{v} = RQ\).

C

1.6.10. It can be shown that any four noncoplanar points (i.e. points that do not lie in the same plane) determine a sphere. Find the equation of the sphere that passes through the points \((0,0,0),\, (0,0,2),\, (1,−4,3) \text{ and }(0,−1,3)\). (Hint: Equation (1.31))

1.6.11. Show that the hyperboloid of one sheet is a doubly ruled surface, i.e. each point on the surface is on two lines lying entirely on the surface. (Hint: Write equation (1.35) as \(\frac{x^ 2}{ a^ 2} − \frac{z^ 2}{ c^ 2} = 1− \frac{y^ 2}{ b^ 2}\) , factor each side. Recall that two planes intersect in a line.)

1.6.12. Show that the hyperbolic paraboloid is a doubly ruled surface. (Hint: Exercise 11)

Let \(S\) be the sphere with radius 1 centered at \((0,0,1)\), and let \(S^∗\) be \(S\) without the “north pole” point \((0,0,2)\). Let \((a,b, c)\) be an arbitrary point on \(S^∗\) . Then the line passing through \((0,0,2) \text{ and }(a,b, c)\) intersects the \(x y\)-plane at some point \((x, y,0)\), as in Figure 1.6.10. Find this point \((x, y,0)\) in terms of \(a, b \text{ and }c\).

Figure 1.6.10

(Note: Every point in the \(x y\)-plane can be matched with a point on \(S^ ∗\) , and vice versa, in this manner. This method is called stereographic projection, which essentially identifies all of \(\mathbb{R}^ 2\) with a “punctured” sphere.)

C

1.7.10. Let \(P = (a,θ,φ)\) be a point in spherical coordinates, with \(a > 0 \text{ and }0 < φ < π\). Then \(P\) lies on the sphere \(ρ = a\). Since \(0 < φ < π\), the line segment from the origin to \(P\) can be extended to intersect the cylinder given by \(r = a\) (in cylindrical coordinates). Find the cylindrical coordinates of that point of intersection.

1.7.11. Let \(P_1 \text{ and }P_2\) be points whose spherical coordinates are \((ρ_1 ,θ_1 ,φ_1) \text{ and }(ρ_2 ,θ_2 ,φ_2)\), respectively. Let \(\mathbf{v}_1\) be the vector from the origin to \(P_1\) , and let \(\mathbf{v}_2\) be the vector from the origin to \(P_2\) . For the angle \(γ\) between \(\mathbf{v}_1 \text{ and }\mathbf{v}_2\) , show that

\[\cos γ = \cos φ_1 \cos φ_2 +\sin φ_1 \sin φ_2 \cos (θ_2 −θ_1 ).\]

This formula is used in electrodynamics to prove the addition theorem for spherical harmonics, which provides a general expression for the electrostatic potential at a point due to a unit charge. See pp. 100-102 in JACKSON.

1.7.12. Show that the distance d between the points \(P_1 \text{ and }P_2\) with cylindrical coordinates \((r_1 ,θ_1 , z_1) \text{ and }(r_2 ,θ_2 , z_2)\), respectively, is

1.8.11. Let a particle of (constant) mass m have position vector \(\textbf{r}(t)\), velocity \(\textbf{v}(t)\), acceleration \(\textbf{a}(t)\) and momentum \(\textbf{p}(t)\) at time \(t\). The angular momentum \(\textbf{L}(t)\) of the particle with respect to the origin at time \(t\) is defined as \(\textbf{L}(t) = \textbf{r}(t)× \textbf{p}(t)\). If \(\textbf{F}(t)\) is the force acting on the particle at time \(t\), then define the torque \(\textbf{N}(t)\) acting on the particle with respect to the origin as \(\textbf{N}(t) = \textbf{r}(t)× \textbf{F}(t)\). Show that \(\textbf{L} ′ (t) = \textbf{N}(t)\).

1.8.13. The Mean Value Theorem does not hold for vector-valued functions: Show that for \(\textbf{f}(t) = (\cos t,\sin t,t)\), there is no \(t\) in the interval \((0,2π)\) such that

\[\textbf{f} ′ (t) = \frac{\textbf{f}(2π)−\textbf{f}(0)}{ 2π−0} .\]

C

1.8.14. The Bézier curve \(\textbf{b}_0^3 (t)\) for four noncollinear points \(\textbf{b}_0 ,\, \textbf{b}_1 ,\, \textbf{b}_2 ,\, \textbf{b}_3\) in \(\mathbb{R}^ 3\) is defined by the following algorithm (going from the left column to the right):

1.8.16. Let \(\textbf{r}(t)\) be the position vector in \(\mathbb{R}^ 3\) for a particle that moves with constant speed \(c > 0\) in a circle of radius \(a > 0\) in the \(x y\)-plane. Show that \(\textbf{a}(t)\) points in the opposite direction as \(\textbf{r}(t) \text{ for all }t\). (Hint: Use Example 1.37 to show that \(\textbf{r}(t) ⊥ \textbf{v}(t) \text{ and }\textbf{a}(t) ⊥ \textbf{v}(t)\), and hence \(\textbf{a}(t) ∥ \textbf{r}(t).\))

Note: The vectors \(\textbf{T}(t), \,\textbf{N}(t) \text{ and }\textbf{B}(t)\) form a right-handed system of mutually perpendicular unit vectors (called orthonormal vectors) at each point on the curve \(\textbf{f}(t)\).